Lattice-valued Borel measures. III.

summary:Let $X$ be a completely regular $T_{1}$ space, $E$ a boundedly complete vector lattice, $ C(X)$ $(C_{b}(X))$ the space of all (all, bounded), real-valued continuous functions on $X$. In order convergence, we consider $E$-valued, order-bounded, $\sigma $-additive, $\tau $-additive, and tight measures on X and prove some order-theoretic and topological properties of these measures. Also for an order-bounded, $E$-valued (for some special $E$) linear map on $C(X)$, a measure representation result is proved. In case $E_{n}^{*}$ separates the points of $E$, an Alexanderov’s type theorem is proved for a sequence of $\sigma $-additive measures